Quotient of Gaussian Distributions has Cauchy Distribution

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Theorem

Let $X$ and $Y$ be independent continuous random variables each with a Gaussian distribution with zero expectation:

\(\ds X\) \(\sim\) \(\ds \Gaussian 0 { {\sigma_x}^2}\)
\(\ds Y\) \(\sim\) \(\ds \Gaussian 0 { {\sigma_y}^2}\)

Let $U$ be the continuous random variable defined as:

$U = \dfrac X Y$


Then $U$ has the Cauchy distribution:

$U \sim \Cauchy 0 \lambda$

where:

$\lambda = \dfrac {\sigma_x} {\sigma_y}$


Corollary

Let $X$ and $Y$ be independent continuous random variables each with a Gaussian distribution with:

zero expectation
the same variance $\sigma$:
\(\ds X\) \(\sim\) \(\ds \Gaussian 0 {\sigma^2}\)
\(\ds Y\) \(\sim\) \(\ds \Gaussian 0 {\sigma^2}\)

Let $U$ be the continuous random variable defined as:

$U = \dfrac X Y$


Then $U$ has the Cauchy distribution:

$U \sim \Cauchy 0 1$

and so does $\dfrac 1 U$:

$\dfrac 1 U \sim \Cauchy 0 1$


Proof


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Sources

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